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A386442 Decimal expansion of Sum_{k>=2} H(k) * (zeta(k) - 1), where H(k) is the k-th harmonic number.

%I A386442 #5 Jul 22 2025 04:31:48
%S A386442 1,6,8,0,5,3,1,2,2,2,0,4,2,8,3,6,7,6,9,4,0,3,3,8,7,7,4,0,4,1,3,4,7,9,
%T A386442 0,9,7,4,6,9,3,8,1,5,5,4,5,6,8,9,6,1,2,7,0,1,7,1,7,7,3,5,9,8,6,3,7,6,
%U A386442 8,2,2,6,8,1,1,6,0,8,0,2,6,4,0,3,3,8,4,3,4,0,9,8,5,8,4,4,4,2,2,8,7,3,0,8,5
%N A386442 Decimal expansion of Sum_{k>=2} H(k) * (zeta(k) - 1), where H(k) is the k-th harmonic number.
%H A386442 Khristo N. Boyadzhiev, <a href="https://arxiv.org/abs/1903.11141">A special constant and series with zeta values and harmonic numbers</a>, arXiv:1903.11141 [math.NT], 2019.
%H A386442 Michael I. Shamos, <a href="https://citeseerx.ist.psu.edu/pdf/ae33a269baba5e8b1038e719fb3209e8a00abec5">Shamos's catalog of the real numbers</a>, 2011. See p. 565.
%F A386442 Equals 1 + Sum_{k>=2} (H(k) - 1) * (zeta(k) - 1).
%F A386442 Equals Sum_{k>=2} ((k/(k-1))*log(k/(k-1)) - 1/k) (Shamos, 2011).
%F A386442 Equals 1 + Sum_{k>=2} (-1)^k * zeta(k) / (k*(k-1)) (Shamos, 2011).
%F A386442 Equals M + 1 - gamma, where M = A131688 and gamma = A001620 (Boyadzhiev, 2019).
%e A386442 1.68053122204283676940338774041347909746938155456896...
%t A386442 RealDigits[NIntegrate[HarmonicNumber[x]/x, {x, 0, 1}, WorkingPrecision -> 120] + 1 - EulerGamma][[1]]
%o A386442 (PARI) sumpos(k = 2, (k/(k-1))*log(k/(k-1)) - 1/k)
%Y A386442 Cf. A001620, A001008, A002805, A131688.
%K A386442 nonn,cons
%O A386442 1,2
%A A386442 _Amiram Eldar_, Jul 21 2025